Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge

Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge

A modified Leslie-Gower predator-prey system with two delays is investigated. By choosing τ1 and τ2 as bifurcation parameters, we show that the Hopf bifurcations occur when time delay crosses some critical values. Moreover, we derive the equation describing the flow on the center manifold; then we g...

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Main Authors: Qingsong Liu, Yiping Lin, Jingnan Cao
Format: Article
Language:English
Published: Hindawi Limited 2014-01-01
Series:Computational and Mathematical Methods in Medicine
Online Access:http://dx.doi.org/10.1155/2014/619132
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spelling doaj-52d0b554567044d2921e3f509f8a32902020-11-24T22:25:48ZengHindawi LimitedComputational and Mathematical Methods in Medicine1748-670X1748-67182014-01-01201410.1155/2014/619132619132Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey RefugeQingsong Liu0Yiping Lin1Jingnan Cao2Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650500, ChinaDepartment of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650500, ChinaSchool of Computer Science, Beijing University of Post Telecommunication, Beijing 100876, ChinaA modified Leslie-Gower predator-prey system with two delays is investigated. By choosing τ1 and τ2 as bifurcation parameters, we show that the Hopf bifurcations occur when time delay crosses some critical values. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and chaotic behaviors are observed. Finally, using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of the periodic solutions.http://dx.doi.org/10.1155/2014/619132
collection DOAJ
language English
format Article
sources DOAJ
author Qingsong Liu
Yiping Lin
Jingnan Cao
spellingShingle Qingsong Liu
Yiping Lin
Jingnan Cao
Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge
Computational and Mathematical Methods in Medicine
author_facet Qingsong Liu
Yiping Lin
Jingnan Cao
author_sort Qingsong Liu
title Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge
title_short Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge
title_full Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge
title_fullStr Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge
title_full_unstemmed Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge
title_sort global hopf bifurcation on two-delays leslie-gower predator-prey system with a prey refuge
publisher Hindawi Limited
series Computational and Mathematical Methods in Medicine
issn 1748-670X
1748-6718
publishDate 2014-01-01
description A modified Leslie-Gower predator-prey system with two delays is investigated. By choosing τ1 and τ2 as bifurcation parameters, we show that the Hopf bifurcations occur when time delay crosses some critical values. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and chaotic behaviors are observed. Finally, using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of the periodic solutions.
url http://dx.doi.org/10.1155/2014/619132
work_keys_str_mv AT qingsongliu globalhopfbifurcationontwodelayslesliegowerpredatorpreysystemwithapreyrefuge
AT yipinglin globalhopfbifurcationontwodelayslesliegowerpredatorpreysystemwithapreyrefuge
AT jingnancao globalhopfbifurcationontwodelayslesliegowerpredatorpreysystemwithapreyrefuge
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